On the Transient Behaviour of Fractional M/ M/ ∞ Queues

We study some features of the transient probability distribution of a fractional M/ M/ ∞ queueing system. Such model is constructed as a suitable time-changed birth-death process. The fractional differential-difference problem is studied for the corresponding probability distribution and a fractional partial differential equation is obtained for the generating function. Finally, the interpretation of the system as an actual M/ M/ ∞ queue and as a M/M/1 queue with responsive server is given and some conditioned virtual waiting times are studied